1 Symmetry in Physics Kihyeon Cho January 29, 2009 High Energy Physics Phenomenology

2 Line Symmetry Shape has line symmetry when one half of it is the mirror image of the other half. Symmetry exists all around us and many people see it as being a thing of beauty.

3 Is a butterfly symmetrical?

4 Line Symmetry exists in nature but you may not have noticed.

5 At the beach there are a variety of shells with line symmetry.

6 Under the sea there are also many symmetrical objects such as these crabs and this starfish.

7 Animals that have Line Symmetry Here are a few more great examples of mirror image in the animal kingdom.

8 THESE MASKS HAVE SYMMETRY These masks have a line of symmetry from the forehead to the chin. The human face also has a line of symmetry in the same place.

9 Human Symmetry The 'Proportions of Man' is a famous work of art by Leonardo da Vinci that shows the symmetry of the human form.

10 REFLECTION IN WATER If an object is reflected in water it is considered to have line symmetry along the waterline.

11 The Taj Mahal Symmetry exists in architecture all around the world. The best known example of this is the Taj Mahal.

12 This photograph shows 2 lines of symmetry. One vertical, the other along the waterline. (Notice how the prayer towers, called minarets, are reflected in the water and side to side).

13 2D Shapes and Symmetry After investigating the following shapes by cutting and folding, we found:

14 an equilateral triangle has 3 internal angles and 3 lines of symmetry.

15 a square has 4 internal angles and 4 lines of symmetry.

16 a regular pentagon has 5 internal angles and 5 lines of symmetry.

17 a regular hexagon has 6 internal angles and 6 lines of symmetry.

18 a regular octagon has 8 internal angles and 8 lines of symmetry.

19

20 Conserved Quantities and Symmetries Every conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation. We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuous Continuous  give additive conservation laws x  x+dx or   +d  examples of conserved quantities: electric charge momentum baryon # Discontinuous  give multiplicative conservation laws parity transformation: x, y, z  (-x), (-y), (-z) charge conjugation (particle  antiparticle): e -  e + examples of conserved quantities: parity (in strong and EM) charge conjugation (in strong and EM) parity and charge conjugation (strong, EM, almost always in weak)

21 quark  anti quark  quark  anti quark ……  decay Time We are all the children of Broken symmetry Just tiny deviation from perfect symmetry seems to have been enough

22 Why matter dominant world? Baryon number violation CP violation Start from thermal equilibrium

23 Three Important Discrete Symmetries Parity, P –Parity reflects a system through the origin. Converts right-handed coordinate systems to left-handed ones. –Vectors change sign but axial vectors remain unchanged x   xL  L Charge Conjugation, C –Charge conjugation turns a particle into its anti-particle e   e  K   K   Time Reversal, T –Changes the direction of motion of particles in time t  t CPT theorem –One of the most important and generally valid theorems in quantum field theory. –All interactions are invariant under combined C, P and T transformations. –Implies particle and anti-particle have equal masses and lifetimes  

24 C, P, T violation? Since early universe… “Alice effect” Intuitively… Boltzmann and S=kln  C is violated P is violated T is violated

25 Parity Quantum Number

26 Charge-conjugation Quantum Number

27 Charge conjugate and Parity CP is the product of two symmetries: C for charge conjugation, which transforms a particle into its antiparticle, and P for parity, which creates the mirror image of a physical system.symmetriescharge conjugationantiparticleparity C

28 C P CP

29 C and P violation! Experiments show that only circled ones exist in Nature C and P are both maximally violated! But, CP and T seems to be conserved, or is it? CP We can test this in 1 st generation meson system: Pions

30 Mirror symmetry  Parity P All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world Charge symmetry  Charge C Particles should behave exactly like their alter egos, antiparticles, which have exactly the same properties but the opposite charge Time symmetry  Time T Physical events at the micro level should be equally independent whether they occur forwards or backwards in time. There are 3 different principles of symmetry in the basic theory for elementary particles Three principles of symmetry Violation in

31 Mirror symmetry  Parity P All events should occur in exactly the same way whether they are seen directly or in mirror. There should not be any difference between left and right and nobody should be able to decide whether they are in their own world or in a looking glass world Charge symmetry  Charge C Particles should behave exactly like their alter egos, antiparticles, which have exactly the same properties but the opposite charge Time symmetry  Time T Physical events at the micro level should be equally independent whether they occur forwards or backwards in time. There are 3 different principles of symmetry in the basic theory for elementary particles Three principles of symmetry Violation in Violation in

32 CP and T violation! For 37 years, CP violation involve Kaons only! Is CP violation a general property of the SM or is it simply an accident to the Kaons only? CP violation T violation K 0   +   - We can test this in 2 nd generation meson system: Kaons Need 3 rd generation system: B-mesons and B-factories

33 노벨 물리학상 2008 Citation: 5483

34 표준모형 (Standard Model) What does world made of? – 6 quarks u, d, c, s, t, b Meson (q qbar) Baryon (qqq) – 6 leptons e, muon, tau e, , 

35 V ub

36 The CKM Matrix Unitary with 9*2 numbers  4 independent parameters Many ways to write down matrix in terms of these parameters

37 F. Di Lodovico, ICHEP 2008

38 Heavy Flavor Physics 정밀측정으로 표준모형의 검증 => 새로운 물리 현상 Foundation New Physics

39 CP and T violation! For 37 years, CP violation involve Kaons only! Is CP violation a general property of the SM or is it simply an accident to the Kaons only? CP violation T violation K 0   +   - We can test this in 2 nd generation meson system: Kaons Need 3 rd generation system: B-mesons and B-factories

40 Symmetry violation 1972, Makoto Kobayashi, Toshihide Maskawa found that why this symmetry was broken in 3X3 matrix,so called CKM matrix quark  anti quark  quark  anti quark ……  decay Time If this exchange of identity which double broken symmetry was to take place between matter and antimatter, a further quark family was needed 1974, c quark J/  1977, b quark  1995, t quark 2008

41 Conserved Quantities and Symmetries Example of classical mechanics and momentum conservation. In general a system can be described by the following Hamiltonian: H=H(p i,q i,t) with p i =momentum coordinate, q i =space coordinate, t=time Consider the variation of H due to a translation q i only. For our example dp i =dt=0 so we have: Using Hamilton’s canonical equations: We can rewrite dH as: If H is invariant under a translation (dq) then by definition we must have: This can only be true if: Thus each p component is constant in time and momentum is conserved.

42 Conserved Quantities and Quantum Mechanics In quantum mechanics quantities whose operators commute with the Hamiltonian are conserved. Recall: the expectation value of an operator Q is: How does change with time? Recall Schrodinger’s equation: Substituting the Schrodinger equation into the time derivative of Q gives: H + = H *T = hermitian conjugate of H Since H is hermitian ( H + = H ) we can rewrite the above as: So then is conserved.

43 Conservation of electric charge and gauge invariance Conservation of electric charge:  Q i =  Q f Evidence for conservation of electric charge: Consider reaction e -  v e which violates charge conservation but not lepton number or any other quantum number. If the above transition occurs in nature then we should see x-rays from atomic transitions. The absence of such x-rays leads to the limit:  e > 2x10 22 years There is a connection between charge conservation, gauge invariance, and quantum field theory. Recall Maxwell’s Equations are invariant under a gauge transformation: A Lagrangian that is invariant under a transformation U=e i  is said to be gauge invariant. There are two types of gauge transformations: local:  =  (x,t) global:  =constant, independent of (x,t) 1. Conservation of electric charge is the result of global gauge invariance 2. Photon is massless due to local gauge invariance <= Maxwell eq. Maxwell’s EQs are locally gauge invariant

44 Assessment Global gauge invariance implies the existence of a conserved current, according to Noether’s theorem. Local gauge invariance requires the introduction of massless vector bosons, restricts the form of the interactions of gauge bosons with sources, and generates interactions among the gauge bosons if the symmetry is non-Abelian. - Chris Quigg P63

45 Gauge invariance, Group Theory, and Stuff Consider a transformation (U) that acts on a wave function (  ):  U  Let U be a continuous transformation then U is of the form: U=e i    is an operator. 1. If  is a hermitian operator (  =  *T ) then U is a unitary transformation: U=e i  U + =(e i  ) *T = e -i  *T = e -i   UU + = e i  e -i  =1 Note: U is not a hermitian operator since U  U + 2. In the language of group theory  is said to be the generator of U There are 4 properties that define a group: 1) closure: if A and B are members of the group then so is AB 2) identity: for all members of the set I exists such that IA=A 3) Inverse: the set must contain an inverse for every element in the set AA -1 =I 4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C 3. If  = (  1,  2,  3,..) then the transformation is “Abelian” if: U(  1 )U(  2 ) = U(  2 )U(  1 ) i.e. the operators commute 4. If the operators do not commute then the group is non-Abelian. The transformation with only one  forms the unitary abelian group U(1) The Pauli (spin) matrices generate the non-Abelian group SU(2) S= “special”= unit determinant U=unitary n=dimension (e.g.2)

46 Global Gauge Invariance and Charge Conservation The relativistic Lagrangian for a free electron is:  is the electron field (a 4 component spinor) m is the electrons mass  u = “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy  u  v +  v  u =2g uv  u          t  x  y  z  Let’s apply a global gauge transformation to L By Noether’s Theorem there must be a conserved quantity associated with this symmetry! This Lagrangian gives the Dirac equation:

47 The Dirac Equation on One Page A solution (one of 4, two with +E, two with -E) to the Dirac equation is : The function U is a (two-component) SPINOR and satisfies the following equation: The Dirac equation: Spinors are most commonly used in physics to describe spin 1/2 objects. For example: Spinors also have the property that they change sign under a rotation!

48 E-L equation in 1D Global Gauge Invariance and Charge Conservation We need to find the quantity that is conserved by our symmetry. In general if a Lagrangian density, L=L( ,  x u ) with  a field, is invariant under a transformation we have: For our global gauge transformation we have: Plugging this result into the equation above we get (after some algebra…) The first term is zero by the Euler-Lagrange equation. The second term gives us a continuity equation. Result from field theory

49 Global Gauge Invariance and Charge Conservation The continuity equation is: Recall that in classical E&M the (charge/current) continuity equation has the form: Also, recall that the Schrodinger equation give a conserved (probability) current: If we use the Dirac Lagrangian in the above equation for L we find: This is just the relativistic electromagnetic current density for an electron. The electric charge is just the zeroth component of the 4-vector: Therefore, if there are no current sources or sinks (  J=0) charge is conserved as: (J 0, J 1, J 2, J 3 ) =( , J x, J y, J z ) Result from quantum field theory Conserved quantity

50 Local Gauge Invariance and Physics Some consequences of local gauge invariance: a) For QED local gauge invariance implies that the photon is massless. b) In theories with local gauge invariance a conserved quantum number implies a long range field. e.g. electric and magnetic field However, there are other quantum numbers that are similar to electric charge (e.g. lepton number, baryon number) that don’t seem to have a long range force associated with them! Perhaps these are not exact symmetries!  evidence for neutrino oscillation implies lepton number violation. c) Theories with local gauge invariance can be renormalizable, i.e. can use perturbation theory to calculate decay rates, cross sections, etc. Strong, Weak and EM theories are described by local gauge theories. U(1) local gauge invariance first discussed by Weyl in 1919 SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)  e i  (x,t)   is represented by the 2x2 Pauli matrices (non-Abelian) SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s  e i  (x,t)   is represented by the 3x3 matrices of SU(3) (non-Abelian)

51 Local Gauge Invariance Strong, Weak and EM theories are described by local gauge theories. U(1) local gauge invariance first discussed by Weyl in 1919 SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak)  e i  (x,t)   is represented by the 2x2 Pauli matrices (non- Abelian) SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s  e i  (x,t)   is represented by the 3x3 matrices of SU(3) (non-Abelian

52 Local Gauge Invariance and QED Consider the case of local gauge invariance, = (x,t) with transformation: The relativistic Lagrangian for a free electron is NOT invariant under this transformation. The derivative in the Lagrangian introduces an extra term: We can MAKE a Lagrangian that is locally gauge invariant by adding an extra piece to the free electron Lagrangian that will cancel the derivative term. We need to add a vector field A u which transforms under a gauge transformation as: A u  A u +  u  (x,t) with (x,t)=-q  (x,t) (for electron q=-|e|) The new, locally gauge invariant Lagrangian is:

53 The Locally Gauge Invariance QED Lagrangian Several important things to note about the above Lagrangian: 1) A u is the field associated with the photon. 2) The mass of the photon must be zero or else there would be a term in the Lagrangian of the form: m  A u A u However, A u A u is not gauge invariant! 3) F uv =  u A v -  v A u and represents the kinetic energy term of the photon. 4) The photon and electron interact via the last term in the Lagrangian. This is sometimes called a current interaction since: In order to do QED calculations we apply perturbation theory (via Feynman diagrams) to J u A u term. 5) The symmetry group involved here is unitary and has one parameter  U(1) e-e- e-e-  AuAu JuJu

54 References I.S.Cho’s talk (2008) Class P by Richard Kass (2003) B.G Cheon’s Summer School (2002) S.H Yang’s Colloquium (2001) Class by Jungil Lee (2004) PDG home page (